Upper Bound Analysis for S

A detailed step‐by‐step method for finding the upper bound

Key Highlights

Measurement Rounding: Understand that values given to 2 significant figures imply an inherent uncertainty which defines an interval of possible actual values.
Determining Bounds: The upper bound for a measurement is calculated by adding half the unit in the last significant digit, while the lower bound is found by subtracting the same amount.
Application to S: When S is defined as the difference between two measured quantities, the maximum (upper bound) for S is obtained by taking the lower bound of the quantity being subtracted from and the upper bound of the quantity from which it is subtracted.

Understanding the Problem

In this experiment, two quantities—denoted D and another (often labeled as T or a constant value)—are measured to 2 significant figures. The problem statement presents a measured value of 7.6 for one element and a value 70.23 for the other.

There can be some ambiguity when interpreting the statement “D-7.6 and 70.23”. One standard interpretation (which we will follow here) is that one part of the measurement (or perhaps a fixed subtraction component) is 7.6, while the other measurement is 70.23. In our analysis, we assume the quantity S is determined by the difference between a measured value (which we interpret as 70.23) and a value 7.6. That is, we define:

Definition of S

\( S = \text{Measured Value} - 7.6 \)

Here the measured value is given as 70.23 to 2 significant figures. Due to the nature of significant figures, the value 70.23 is understood as an approximation rounded to 2 significant figures. This rounding establishes an interval within which the actual value lies.

Calculating Upper and Lower Bounds

For a measured figure correct to 2 significant figures, the precision is determined by the position of the last significant digit. The half-width (or half of the unit in the last significant place) sets the tolerance in both directions.

Step 1 – Analyzing the Constant 7.6

The value 7.6 has its last significant digit in the tenths place. The unit in this place is 0.1. Therefore, the half interval is:

\( \frac{0.1}{2} = 0.05 \)

This means:

Upper bound for 7.6 = \( 7.6 + 0.05 = 7.65 \)
Lower bound for 7.6 = \( 7.6 - 0.05 = 7.55 \)

Step 2 – Analyzing the Measured Value 70.23

Although 70.23 appears to have more digits, it is stated that it is measured correct to 2 significant figures. This means the true measurement was recorded to only 2 significant digits; the extra digits are a result of rounding. When a measurement is quoted to 2 significant figures, the value is usually expressed in a form that might mask the true precision. For example:

70.23 rounded to 2 significant figures gives 70. However, when considering bounds we must recover the interval that a value “70” (to 2 significant figures) represents.

The procedure is as follows:

Determine the rounding unit: for a number like 70 (since it is 2 significant figures and the first digit is in the tens place), the appropriate unit is 10. However, in many standard treatments (especially within certain contexts such as experimental measurements), it is acceptable in some cases to interpret a value such as 70.23 with a display of extra decimals as a more precise result. In our analysis, we adopt the interpretation where the inherent measurement uncertainty is based on the position of the first nonzero digit. Here, we assume that the intended precision corresponds to the tens place.
For example, if 70 is the rounded value based on 2 s.f, then the half-interval is found from the decimal place corresponding to the significant figures. In this context, a common approach is to use an interval of 1 (if the rounding were to the nearest 10, half the interval would be 5).

Thus, for a value rounded to 2 significant figures represented as 70:

Upper bound for the measured value = \( 70 + 0.5 = 70.5 \)
Lower bound for the measured value = \( 70 - 0.5 = 69.5 \)

Note that if additional context suggests that the extra digits (“.23”) carry meaning (as in the case where a leading measurement is recorded with extra precision despite being marked as 2 s.f.), then one might be tempted to treat it with a different unit. However, standard convention for 2 significant figures confirms the bounds as calculated above.

Step 3 – Establishing S Upper Bound

For a difference S, calculated as:

\( S = (\text{Measured Value}) - 7.6 \)

The maximum possible (upper bound) for S is obtained by:

Maximizing the minuend (the measured value).
Minimizing the subtrahend (the value 7.6) as much as possible.

Given our bounds:

Upper bound of the measured value: 70.5
Lower bound for 7.6: 7.55 (using the lower bound because subtracting a smaller number gives a larger difference)

Therefore, the upper bound for S will be:

Calculation:

\( S_{\text{upper}} = 70.5 - 7.55 \)

Evaluating:

70.5 − 7.55 = 62.95

This gives an upper bound for S of approximately 62.95.

Detailed Analysis Table

The following table summarizes the bounds for each measurement and the resulting calculation for S:

Measurement	Recorded Value	Rounding Unit	Half-Interval	Lower Bound	Upper Bound
Constant (Subtrahend)	7.6	0.1	0.05	7.55	7.65
Measured Value (Minuend)	70 (interpreted from 70.23 rounded to 2 s.f.)	10	5	69.5	70.5

Note: Although the recorded value is shown as 70.23, when strictly interpreted to 2 significant figures it equates to 70, and hence the corresponding bounds are 69.5 and 70.5.

Final Calculation and Interpretation

Using the bounds derived above:

Upper Bound for S

\( S_{\text{upper}} = (\text{Upper Bound of Measured Value}) - (\text{Lower Bound of Constant}) \)

\( S_{\text{upper}} = 70.5 - 7.55 = 62.95 \)

In many contexts – particularly when small measurement uncertainties exist – it is acceptable to round the final upper bound to a sensible number of decimals. In this case, one might express this as approximately 63.

Hence, the upper bound for S is approximately 63.

Conclusion

The analysis involves understanding how significant figures affect the measurement bounds of the recorded values. For the constant 7.6 measured to 2 significant figures, the bounds are 7.55 to 7.65. For a value that is recorded as 70.23 but understood to be 70 when rounded to 2 significant figures, the appropriate bounds become 69.5 to 70.5. When calculating S defined as the difference between the measured value and the constant, its upper bound is achieved by combining the maximum possible value of the measured quantity (70.5) with the minimum possible value of the constant (7.55), yielding:

\( S_{\text{upper}} = 70.5 - 7.55 \approx 62.95 \)

Rounded suitably, the upper bound for S is approximately 63.

In conclusion, using the principles of upper and lower bounds in measurements, we establish that the upper bound for S is about 63. This result underscores the importance of careful treatment when combining measurements with inherent uncertainties.